*HYPERELASTIC

Keyword type: model definition, material

This option is used to define the hyperelastic properties of a material. There are two optional parameters. The first one defines the model and can take one of the following strings: ARRUDA-BOYCE, MOONEY-RIVLIN, NEO HOOKE, OGDEN, POLYNOMIAL, REDUCED POLYNOMIAL or YEOH. The second parameter N makes sense for the OGDEN, POLYNOMIAL and REDUCED POLYMIAL model only, and determines the order of the strain energy potential. Default is the POLYNOMIAL model with N=1. All constants may be temperature dependent.

Let $\bar{I}_1$,$\bar{I}_2$ and $J$ be defined by:

$$\bar{I}_1 = III_C^{-1/3} I_C$$ (173)

$$\bar{I}_2 = III_C^{-1/3} II_C$$ (174)

$$J = III_C^{1/2}$$ (175)

where $I_C$, $II_C$ and $III_C$ are the invariants of the right Cauchy-Green deformation tensor $C_{KL}=x_{k,K}x_{k,L}$. The tensor $C_{KL}$ is linked to the Lagrange strain tensor $E_{KL}$ by:

$$2E_{KL}=C_{KL}-\delta_{KL}$$ (176)

where $\delta$ is the Kronecker symbol.

The Arruda-Boyce strain energy potential takes the form:

$$U = \mu \Bigg\{ \frac{1}{2}(\bar{I}_1-3)+\frac{1}{20\lambda_m^2}(\bar{I}_1^2-9)+\frac{11}{1050\lambda_m^4}(\bar{I}_1^3-27)$$

$$+ \frac{19}{7000\lambda_m^6}(\bar{I}_1^4-81)+\frac{519}{673750\lambda_m^8}(\bar{I}_1^5-243) \Bigg\}$$ (177)

$$+ \frac{1}{D} \left( \frac{J^2-1}{2} - \ln J \right)$$

The Mooney-Rivlin strain energy potential takes the form:

$$U=C_{10}(\bar{I}_1-3)+C_{01}(\bar{I}_2-3)+\frac{1}{D_1}(J-1)^2$$ (178)

The Mooney-Rivlin strain energy potential is identical to the polynomial strain energy potential for $N=1$.

The Neo-Hooke strain energy potential takes the form:

$$U=C_{10}(\bar{I}_1-3)+\frac{1}{D_1}(J-1)^2$$ (179)

The Neo-Hooke strain energy potential is identical to the reduced polynomial strain energy potential for $N=1$.

The polynomial strain energy potential takes the form:

$$U=\sum_{i+j=1}^{N} C_{ij}(\bar{I}_1-3)^i(\bar{I}_2-3)^j +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}$$ (180)

In CalculiX $N\le 3$.

The reduced polynomial strain energy potential takes the form:

$$U=\sum_{i=1}^{N} C_{i0}(\bar{I}_1-3)^i +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}$$ (181)

In CalculiX $N\le 3$. The reduced polynomial strain energy potential can be viewed as a special case of the polynomial strain energy potential

The Yeoh strain energy potential is nothing else but the reduced polynomial strain energy potential for $N=3$.

Denoting the principal stretches by $\lambda_1$, $\lambda_2$ and $\lambda_3$ ( $\lambda_1^2$, $\lambda_2^2$ and $\lambda_3^2$ are the eigenvalues of the right Cauchy-Green deformation tensor) and the deviatoric stretches by $\bar{\lambda}_1$, $\bar{\lambda}_2$ and $\bar{\lambda}_3$, where $\bar{\lambda}_i=III_C^{-1/6}\lambda_i$, the Ogden strain energy potential takes the form:

$$U=\sum_{i=1}^{N} \frac{2 \mu_i}{\alpha_i^2}(\bar{\lambda}_1^{\alp... ...{\alpha_i}+\bar{\lambda}_3^{\alpha_i}-3)+\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}.$$ (182)

The input deck for a hyperelastic material looks as follows:

First line:

• *HYPERELASTIC
• Enter parameters and their values, if needed

Following line for the ARRUDA-BOYCE model:

• $\mu$.
• $\lambda_{m}$.
• $D$.
• Temperature

Repeat this line if needed to define complete temperature dependence.

Following line for the MOONEY-RIVLIN model:

• $C_{10}$.
• $C_{01}$.
• $D_{1}$.
• Temperature

Repeat this line if needed to define complete temperature dependence.

Following line for the NEO HOOKE model:

• $C_{10}$.
• $D_{1}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the OGDEN model with N=1:

• $\mu_{1}$.
• $\alpha_{1}$.
• $D_{1}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the OGDEN model with N=2:

• $\mu_{1}$.
• $\alpha_{1}$.
• $\mu_{2}$.
• $\alpha_{2}$.
• $D_{1}$.
• $D_{2}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence.

Following lines, in a pair, for the OGDEN model with N=3: First line of pair:

• $\mu_{1}$.
• $\alpha_{1}$.
• $\mu_{2}$.
• $\alpha_{2}$.
• $\mu_{3}$.
• $\alpha_{3}$.
• $D_{1}$.
• $D_{2}$.

Second line of pair:

• $D_{3}$.
• Temperature.

Repeat this pair if needed to define complete temperature dependence.

Following line for the POLYNOMIAL model with N=1:

• $C_{10}$.
• $C_{01}$.
• $D_{1}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the POLYNOMIAL model with N=2:

• $C_{10}$.
• $C_{01}$.
• $C_{20}$.
• $C_{11}$.
• $C_{02}$.
• $D_{1}$.
• $D_{2}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence.

Following lines, in a pair, for the POLYNOMIAL model with N=3: First line of pair:

• $C_{10}$.
• $C_{01}$.
• $C_{20}$.
• $C_{11}$.
• $C_{02}$.
• $C_{30}$.
• $C_{21}$.
• $C_{12}$.

Second line of pair:

• $C_{03}$.
• $D_{1}$.
• $D_{2}$.
• $D_{3}$.
• Temperature.

Repeat this pair if needed to define complete temperature dependence.

Following line for the REDUCED POLYNOMIAL model with N=1:

• $C_{10}$.
• $D_{1}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the REDUCED POLYNOMIAL model with N=2:

• $C_{10}$.
• $C_{20}$.
• $D_{1}$.
• $D_{2}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the REDUCED POLYNOMIAL model with N=3:

• $C_{10}$.
• $C_{20}$.
• $C_{30}$.
• $D_{1}$.
• $D_{2}$.
• $D_{3}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the YEOH model:

• $C_{10}$.
• $C_{20}$.
• $C_{30}$.
• $D_{1}$.
• $D_{2}$.
• $D_{3}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence.

Example:

*HYPERELASTIC,OGDEN,N=1
3.488,2.163,0.

defines an ogden material with one term: $\mu_{1}$ = 3.488, $\alpha_{1}$ = 2.163, $D_{1}$=0. Since the compressibility coefficient was chosen to be zero, it will be replaced by CalculiX by a small value to ensure some compressibility to guarantee convergence (cfr. page ).

Example files: beamnh, beamog.